An alternate point-wise scheme for Electric Field Integral Equations

Transcription

1 An alternate pont-wse scheme for Electrc Feld Integral Equatons Gabrele Rosat, Juan R. Mosg Laboratory of Electromagnetcs and Acoustcs (LEMA) Ecole Polytechnque Fédérale de Lausanne (EPFL), CH-05 Lausanne, wtzerland Receved*************** ; accepted ************** ABTRACT: A pont-wse approach that can be used effcently n the numercal soluton of Electrc Feld Integral Equatons s ntroduced. The algorthm s based on the so-called magc dstance concept, whch defnes exactly the pont-to-pont equvalent of a four dmensonal ntegral. Magc dstance values are rgorously obtaned n the electrostatc case and ther usefulness s demonstrated. The concept s generalzed to the electrodynamc case, resultng n a famly of very smple "magc-dstance nspred" algorthms that mantan the overall accuracy of Galerkn formulatons wthout apprecable deteroraton n the overall accuracy. 203 Wley Perodcals, Inc. Int J RF and Mcrowave CAE xx:xxx-xxx, 203 Keywords:Electrc Feld Integral Equaton (EFIE), method of moments (MoM), Galerkn, pontmatchng Correspondence to: G. Rosat; emal: gabrele.rosat@epfl.ch DOI. Publshed onlne... n Wley Onlne Lbrary (wleyonlnelbrary.com)

2 Rosat and Mosg I. INTRODUCTION Integral Equaton (IE) formulatons are one of the most successful technques for the EM analyss of antennas and scatterers []. Ths s specally true when the nvolved obects can be reduced to combnatons of metallc surfaces and volumes embedded n a 3D homogeneous or multlayered medum [2]. The classcal dscretzaton scheme and subsequent numercal mplementaton of IE formulatons s usually performed wth the help of the method of moments (MoM). In partcular, the IE-MoM mplementaton known as Galerkn method s endowed wth varatonal propertes, whch guarantees good convergence propertes [3]. The most commonly used IE formulaton s the Electrc Feld Integral Equaton (EFIE). On one sde t nvolves only the tangental component of an unquely defned physcal quantty, the electrc feld. In addton, t can be easly appled to metallc open surfaces that are so ubqutous n modern technologes. Wthn the EFIE famly, perhaps the most useful verson s the Mxed Potental Integral Equaton (MPIE). As t s well known [,4], MPIE exhbts source-observer sngulartes much mlder (/R^) than the orgnal EFIE (/R^3) and allows the use of smpler MoM schemes lke pontmatchng. But as ts name says, MPIE s based on the ntroducton of auxlary mathematcal quanttes, the potentals, whose defnton, exstence and unqueness are frequently hard to establsh n complex meda. Ths paper ntroduces some nnovatve very smple pont-wse MoM schemes amng at reducng the numercal complexty of the Galerkn method wthout producng a szeable deteroraton n the accuracy of the results. The developments are based n the ntroducton of a new concept, the so-called "magc dstance". Ths concept wll be ntroduced n a smplfed electrostatc case and then mplemented n the frame of EFIE and of ts weaker sngular verson, the MPIE. The mplementaton remans at a proof-of-concept level usng free space Green's functons and a smplfed mesh, snce the extenson to more complex stuatons should not requre any addtonal qualtatve development. II. THE INTEGRAL EQUATION As mentoned n the ntroducton, we wll consder open metallc surfaces embedded n a 3D envronment and we wll concentrate here n the Electrc Feld Integral Equaton (EFIE). EFIE only nvolves tangental electrc felds, whch are always mathematcally welldefned quanttes, snce they have an obvous physcal meanng. In electrodynamcs, EFIE can be wrtten as [,4]: nˆ G (, ) ( ) ˆ s = ( ) EJ r r J r n E r (a) r r where s the combnaton of all metallc surfaces, E s the ncdent exctaton feld, J s s the unknown surface current densty and G EJ s the dyadc Green's functon, kernel of the ntegral equaton. Ths Green's functon wll usually nclude the effect of the materal envronment surroundng the metallc sheets. Therefore, the complexty of the dyadc GF goes from the closed-from analytcal formulaton for free space to qute nvolved expressons, lke the combnatons of ommerfeld ntegrals and analytcal seres found n the treatment of planar, cylndrcal and sphercal layered delectrc meda [5]. However, n all cases, the EFIE GF ntrnsc sourceobserver sngularty s of /R^3 type, where R = r r s the source-observer dstance [,2]. Ths s obvous n the well-known analytcal form of the dyadc Green's functon for free space []: ( kr) ωμ exp GEJ ( r, r ) = I + 4π k 2 (b) R Ths means that when dscretzng the EFIE wth a classcal Method of Moments (MoM) procedure, a smple scheme lke pont-matchng cannot be used. Ths strong sngularty also prevents the use of otherwse promsng approaches lke the pont-wse Nyström method, based on numercal quadrature rules [6,7]. Ths spatal sngularty problem can be avoded by solvng the EFIE n the spectral doman [,4]. However, spectral approaches rely heavly n the extensve use of 2D Fourer-transforms and are neffcent for arbtrary or non-canoncal geometres. In practce, successful mplementatons of EFIE (FEKO, WIPL-D) reman n the space doman and resort to the use of subsectonal bass and a MoM- Galerkn approach. The Galerkn mplementaton of the EFIE results then n matrces whose elements are gven by double surface (and hence 4D) ntegrals, where - at least n theory- an /R^3 sngularty can be ntegrated. A more popular strategy s to transform the EFIE nto a mxed potental ntegral equaton (MPIE). Electromagnetc theory allows us to derve the electrc feld from two potentals n a classcal way E= ωa V (2) Usng ths equvalence, EFIE can be transformed nto MPIE: A (, ) s ( ) ( rr, ) ρ ( r) ω ds + ˆ G r r J r n = nˆ E r ds G V s r ( ) where n addton to the unknown surface current densty unknown, the surface charge densty ρ s, also appears. r (3a) J s, a new It s a well-known fact [], that the sngulartes n both the scalar Green's Functon G V for the scalar potental and the dyadc Green's Functon G A for the vector potental exhbt only a mld sourceobserver sngularty of the /R type that can be absorbed n the surface source ntegral. Therefore, smple pont-matchng dscretzaton can be consdered wth MPIE. In partcular, n free space, the use of the classcal Lorentz' gauge [2] results n the followng expressons for the potental Green's functons: exp( kr) GV ( rr, ) = (3b) 4πε R μ exp( kr) GA ( r, r ) = (3c) 4π R Internatonal Journal of RF and Mcrowave Computer-Aded Engneerng

3 As t has been prevously ponted out, the potentals are somewhat artfcal quanttes that can be defned n several ways and connected through dfferent gauge equatons (Coulomb, Lorentz...). Ths also apples to the assocated Green's functons. For nstance, t s well known [8] that n a multlayered meda the value of the scalar Green's functon G V can depend on the orentaton of the orgnal dpole source, a strange stuaton for a scalar potental whch s supposed to be created by a pont charge. In these cases, t s necessary to add a correcton term to the orgnal EFIE expresson (3) [5]. The fnal ntegral equaton of relevance n ths paper s the "electrostatc potental ntegral equaton" (EPIE), vewed here as an useful zero-frequency case of the MPIE formulaton. n ˆ ( ) ( ) ˆ ds GV, ρ s = ( ) r r r n E r r r (4a) where the scalar Green's functon s now smply: G V ( rr, ) = (4b) 4πε R In ths case, a usual subsequent transformaton s to defne an "ncdent" or "external" potental as E ( r) = V and to ntegrate the ntegral equaton between a reference pont defned as "ground" and a pont on the metallc surfaces. The fnal result s: ds GV (, ) ρs( ) + V ( ) = C rr r r r r (4c) whch s the complete formulaton of the EPIE. In a smpler verson, there s no external potental and the constant C s the known potentals of the metallc obects respect to ground. The obvous challenge here s to combne the advantages of EFIE and MPIE,.e. to be able to cast the problem n terms of a physcally well defned quantty lke the electrc feld but avodng the strong sngular EFIE ntegrals. Then, t should be possble to use the pontwse MoM schemes typcal of MPIE, nstead of the cumbersome Galerkn approaches usually need to cancel these strong sngulartes. Therefore, t s of great relevance to nvestgate under whch assumptons pont-matchng or Nyström lke strateges can be drectly appled to the EFIE. Ths s the man purpose of ths paper. III. MOM DICRETIZATION In the standard applcaton of the MoM dscretzaton [D], the unknown quantty s expanded as a lnear combnaton of bass b r and the IE s proected, through an nternal functons ( ) product on a set of test functons t ( ) r. As a result, the ntegral equaton s dscretzed and replaced by a lnear system, where the MoM matrx elements are gven by 4D ntegrals whch can be wrtten as: = ( ) ( ) EJ (, ) z ds t r ds b r G r r (5) for the EFIE and An alternate pont-wse scheme for Electrc Feld Integral Equatons ( ) ( ) A ( ) z= dst r ds b r G r r 2 ds t r ds b r GV r r k 0 for the MPIE. ( ) ( ) ( ) In the Galerkn verson, we smply have: t ( r) = b ( r ) Obvously, the calculaton of 4D ntegrals s cumbersome and requres a lot of computatonal tme and resources. Ths s especally the case for the self-terms (the dagonal terms n the MoM matrx) where there s a need to deal wth sngular and even hypersngular ntegrals [9]. We propose to replace the 4D ntegral requred for each entry of the MoM matrx wth the computaton of a set of pont-to-pont nteractons between an nfntesmal source and some observaton ponts. ource and observaton ponts are arranged n such a confguraton that ensures an exact value for the self-term ( = ) and mnmzes the error n key near-dagonal terms. Once determned, the ponts dstrbuton remans fxed n the calculaton of every cell-to-cell nteracton, thus leadng to a very smple and compact mplementaton. In fact, the proposed approach can be vewed as the development of smple quadrature rules specfcally adapted to a gven type of sngular ntegrals. Moreover, when properly executed, ths strategy should also lead to errors that naturally vansh asymptotcally when the source-observer dstance ncreases wthout bound. It s expected that an exact representaton of the self-term and an accurate approxmaton of the frst near-term(s), whch combne to form the most weghng entres n the MoM matrx, wll mnmze the error n the fnal unknown quanttes, computed through the resoluton of the assocated lnear system. In ths paper, we present dfferent possble confguratons for the electrostatc (ec. IV) and electrodynamc (ec. V) problems. It s shown that t s possble n both cases to fnd a pont confguraton whch allows to approxmate the computaton of the 4D ntegral wth lmted error and to compute the unknown quantty (charge or current) wth very good accuracy. Maor benefts of our method are the smplcty of mplementaton, ts applcablty to problems where potentals cannot be defned, the reduced demand of computatonal tme and resources due to the replacement of 4D ntegrals wth pont-to-pont calculatons. To develop our strategy we wll consder a classcal MoM-Galerkn dscretzaton usng Rao-Wlton-Glsson (RWG) rooftops as bass functons defned on smple geometrcal meshes formed by rectangular cells. Rectangular or square meshes are of nterest by themselves. But n addton, t s always possble to apply the results obtaned for rectangular or square cells to trapezodal/trangular meshes by ntroducng geometrcal affne concepts or usng homothetc changes of varables, exactly lke when defnng twodmensonal quadratures over non-canoncal surfaces. In order to clarfy the above mentoned deas, we start dealng wth the EPIE (electrostatc case). As prevously mentoned, ths case s of paramount relevance when full-wave (electrodynamc) problems are cast n terms of a mxed-potental ntegral equaton, snce then the (6) Internatonal Journal of RF and Mcrowave Computer-Aded Engneerng

4 Rosat and Mosg sngular part of the MPIE self-terms (whch typcally must be ether extracted or separately treated) s gven by the EPIE self-term. IV. THE ELECTROTATIC PROBLEM The electrostatc potental ntegral equaton (EPIE), gven by eqns. 4, doesn't nclude any dfferental operator. Therefore t can be dscretzed wth a Galerkn scheme usng scalar pecewse constant subsectonal bass and test functons, defned as: r b( r) = t( r ) = (7) 0 otherwse The value z self of a generc MoM self-term s then smply gven by ( x x ) 2 + ( y y ) 2 (8) z self = ds ds where s a rectangular cell. Amazngly enough, ths 4D ntegral can be fully computed analytcally. If a,b are the dmensons of the rectangle and c ts dagonal we get: z self = 2( a + b c ) / 3 + 2a 2 bln[( c+ b) / a] + 2ab 2 ln[( c+ a) / b] Now f the calculus mean theorem s appled to the 4D ntegral (8) we should get zself = a b = a b 2 2 Rm < + > ( x x ) ( y y ) (9) (0) In ths last expresson, R m s an "average" or "mean" dstance n the Galerkn sense, obtaned by equatng eqns. 9 and 0. It can be vewed as the dstance between a pont source and a observaton pont whch produces an effect dentcal to the effect of a dstrbuted source averaged on all ts ponts. nce ths dstance embodes n a smple unque pont-to-pont value a complex nteracton descrbed by a four dmensonal ntegral, we can worthly call t a "magc dstance". For the specfc case where s a unt square, the magc dstance takes the nterestng value: R [4ln( 2 ) 4( 2 ) / 3] m = + = () The magc dstance s also defned by the followng remarkable property: f two ponts are randomly selected wthn a gven doman and the dstance between them s computed, the magc dstance s the harmonc average of these dstances. To test the varablty of ths magc dstance concept, we have also computed ts value when s a crcular doman of radus a. Whle t s obvously not a good MoM practce to decompose an arbtrary surface nto crcular domans, the results obtaned for crcular cells are of the uppermost theoretcal relevance and they can provde support to the applcablty of the magc concept dstance to other shapes. The nner surface ntegral n (8) corresponds physcally to the potental created by a constant surface charge densty n any pont ( x, y) of the crcular doman where t s defned. Ths s a classcal problem n Electrostatcs and ts soluton can be wrtten n terms of complete ellptc ntegrals [0]. After some algebrac manpulatons, the Galerkn self-term (8) s gven by a surface ntegral over the observaton doman of some ellptc functons E( t ). More precsely, takng nto account the azmuthal symmetry, we get: 3 z 8 E( ) 6 3 self = πa t tdt = πa /3 (2) 0 Ths s a very nterestng result. To compare t wth the unt square, we consder a unt surface crcle ( π a 2 = ). Then, accordng to (0) the magc dstance for such a crcle s: Rm = 6 / 3 π = (3) a remarkable result when compared wth () whch shows the robustness of the magc dstance concept. The magc dstance concept s a theoretcal dea but t can be easly generalzed to be transformed nto a practcal tool. In the self case, the nner ntegral (assocated wth the source) s replaced wth a pont charge located n the cell's center. The outer ntegral could be replaced by a sngle pont separated from the cell's center by the magc dstance. However, snce we are dealng wth rectangular cells t wll be more symmetrcal to use 4 observer ponts dsposed along the rectangles' medans and all at the magc dstance from the center. In prncple, ths smple arrangement, reproduced n all the cells, wll guarantee perfect results for the dagonal matrx terms and a moderate error n the near-dagonal terms. Moreover, ths error should vansh as we go far away from the dagonal matrx elements correspond n prncple to nteractons between cells separated by larger dstances. More complex dstrbutons of source and/or observer ponts (they can be exchanged by recprocty) wll provde addtonal degrees of freedom allowng the obtenton of exact values n the Z-matrx not only for the dagonal terms but also for other off-dagonal terms. In ths last case, the magc dstance value s used as startng pont for a fast MatLab-based optmzaton. As an example, usng 2 groups of 4 observaton ponts each, dentfed by the dstances d (green) and d2 (red) n fg., t s possble to enforce the matchng condton on the self- and on the frst near-terms. Fgure. Confguraton wth 2 groups of 4 observaton ponts. In all the cases, the generc entry of the Z matrx s therefore calculated as: M z= (4) 2 2 m= ( xm, x) + ( ym, y) Internatonal Journal of RF and Mcrowave Computer-Aded Engneerng

5 both for dagonal and non-dagonal matrx elements. The above qualtatve (and somewhat speculatve) consderatons are now confrmed wth a practcal IE applcaton where we consder a unt square patch above an nfnte ground plane. In a frst run, a full Galerkn approach was compared wth the method usng a sngle cross of four observaton ponts all located at the magc dstance (eqn. ) from the cell's center. The use of the magc dstance produces an error n the MoM matrx whch s maxmum on the frst near-dagonal term and s always less than 6%, as seen n fg. 2a. Once the MoM matrx s nverted the unknown charge surface charge densty can be computed. Its values approxmate very well the "exact" Galerkn soluton (fg. 2b) and the error s on the capactance s around.5%. An alternate pont-wse scheme for Electrc Feld Integral Equatons the dyadc G EJ (eqn. b), whch strongly sngular. In the ntegral equaton (3) the unknown s the surface current J whch s now a vector quantty. Moreover, usng the standard MoM-Galerkn formulaton the bass and test functons are defned by couples of adacent cells rather than by ndvdual cells. Ths shows clearly that the determnaton of a sngle full-wave magc dstance for self-terms s not possble, snce n standard electrodynamc formulatons the dagonal terms ncludes both the nteracton of a cell wth tself and wth the adacent one. Therefore, the smplest scheme s stll nspred by the magc dstance concept but nvolves at least an addtonal degree of freedom, lke the confguraton depcted n fg.. Fnally, the hgher sngulartes n the electrodynamc case call for hgher-order bass and test functons (typcally the surface current densty s expanded wth lnear rooftops rather than constant pulses). Therefore, to obtan a proper pont-wse approxmaton of a Galerkn method, the dfferent source and observaton ponts must be assocated to specfc weghts related to the bass and test functons beng consdered. Ths last statement can be easly clarfed consderng the exact expresson for the generc z term of the MoM matrx: Fgure 2. (a) Detal of the error on the MoM matrx. (b) urface charge. For a more accurate approxmaton, eght observaton ponts located as depcted n fg. are now used on each cell. The addtonal degree of freedom s used to mnmze the error on the closest-to-dagonal terms (nteractons between adacent cells). The error n the MoM matrx elements s dramatcally reduced (fg. 3a), the surface charge s very accurately reproduced (fg. 3b) and the error on the capactance s reduced below 0.5%. The speed-up factor s obvous snce usng the magc dstance strategy the number of elementary computatons needed to evaluate the matrx elements s always the same (four or eght), regardless of the source-observer dstance whle Galerkn wll requre an complex treatment for cancellng the sngularty of the the self-term and at 4 least M elementary computatons The needed order of the quadrature rule, M oscllates typcally between 4 and 6. Fgure 3. (a) Detal of the error on the MoM matrx. (b) urface charge. V. THE ELECTRODYNAMIC FULL WAVE PROBLEM The full-wave problem presents some aspects that dffer from the statc problem and need to be treated. The Green s functon s now = ( ) ( ) EJ (, ) z ds b r ds b r G r r (5) If all the possble source-observer nteractons are reduced to a sngle source pont r 0 and M observaton ponts r m then, the obvous approxmaton for eqn.5 s: M z = M GEJ ( r m,r0) M m (6) m= where ( ) M = ds b r (7) s the total moment of the -th bass functon and Mm = ds b ( ) σ r (8) m are the partal moments correspondng to the partal regons assocated to the m-th observaton pont n the -th cell. σ m Indeed, these partal moments can be consdered as a vectorzed form of the classcal weghts n numercal quadratures. In order to numercally ascertan the valdty and accuracy of these deas a square metallc patch was consdered. As n the statc case, the patch was meshed wth 0 x 0 square cells. The patch lnear sze s one free-space wavelength λ and the exctaton term n the EFIE was provded by a plane wave ( k z, E x ). In the followng developments, the optmum locaton of the testng ponts and the assocated weghts were obtaned as the soluton of a optmzaton problem usng a well defned cost functon and the MATLAB bult-n optmzaton algorthm wth the magc dstance as ntal guess. Internatonal Journal of RF and Mcrowave Computer-Aded Engneerng

6 Rosat and Mosg mple sngle source x(2x4) algorthm The frst logc attempt conssts n applyng a source/observaton pont scheme as n the electrostatc problem,.e. N = source ponts and M = 2x4 observaton ponts, as n fg.. The most crtcal terms after the dagonal self-terms are those correspondng to overlappng bass and test functons that share one cell. Here, two dfferent cases arse, dependng on whether the bass/test functons are parallel or perpendcular. In order to deal wth these two cases, an addtonal degree of freedom s needed. Ths s easly obtaned f the cross' arms do not have necessarly the same length leavng to the modfed confguraton of fg. 4. dagonal terms are exactly computed and the error n the most sgnfcatve MoM matrx elements s below 3% wth an RM error of 6.5%. uch results make meanngful to nvert the matrx to compute the surface current and compare t wth the result of a standard MoM-Galerkn algorthm. Results are shown n Fg. 6(b). The domnant component of the current s obvously the one parallel to the exctaton feld and correspond to bass (9-80) n fg. 6b. Ths component of the current s represented wth satsfyng accuracy: the current peaks are caught and only a small devaton s observed (RM below 0%). The stronger error whch affects the quadrature (y-drected) s less relevant, taken nto account the much lower values of ths current-component. The RM error of around 5% could be tolerated n most practcal applcatons. Fgure 4. Confguraton wth 2 groups of 4 ponts each The weghts assocated to the ponts can be easly obtaned, accordng to (4), assocatng to every test pont angular regons defned by the bsectors of the poston vectors of the test ponts (fg. 4) wth the results (for, respectvely pont n the green x-axs, red, green y-axs): 2 L π tan L 3π L 2 3π cot cot L L π L 3π L 2 3π tan + cot cot Improved sngle source x(3x4) algorthm Followng the prevous descrbed approach, t s straghtforward to ncrease the number of observaton ponts at wll untl reachng a prescrbed accuracy. Fg.5 shows a very good compromse between complexty and accuracy, obtaned wth stll a unque sngle source pont and stll only 3x4 observaton ponts. Fgure 6. (a) Relatve error on the frst row of the MoM matrx and RM error. (b) urface current. Double source (2x6) algorthm The above descrbed algorthms provde qute satsfactory results but they are unnecessarly unbalanced n terms of source and observaton ponts. Indeed, accordng to recprocty theorems, more satsfactory stuatons should be obtaned by usng closer numbers of source and observaton ponts. Extensve numercal trals have shown the amazng benefts obtaned by ust usng N = 2 source ponts and keepng a magc dstance nspred confguraton n the observaton ponts. A fast optmzaton led to the confguraton of fg. 7, usng only M = 6 observaton ponts, thus halvng the computatonal effort wth respect to the prevous confguraton. Fgure 5. Confguraton wth 3 group of 4 ponts each When tested on the one wavelength square plate, ths pont-wse scheme produced a satsfactory MoM matrx. Fg. 6(a) shows the error on the frst row of ths MoM matrx. The dagonal and near Fgure 7. Confguraton of N = 2 source ponts and M = observaton ponts. The error n the MoM matrx entres s shown n Fg. 8(a), and t s remarkable that most elements show now an error below 2% wth the Internatonal Journal of RF and Mcrowave Computer-Aded Engneerng

7 RM error qute small (5%). The sporadc hgher errors n some specfc MoM matrx entres do not affect the overall qualty of the results. Ths s clearly demonstrated n fg 8b where the surface current densty, obtaned after matrx nverson, matches perfectly the MoM-Galerkn values. Indeed, Fg. 9 provdes a qualtatve comparson of the surface currents on the whole patch and confrms the excellence of ths last algorthm. Fgure 8. (a) Relatve error on the frst row of the MoM matrx and RM error. (b) urface current. An alternate pont-wse scheme for Electrc Feld Integral Equatons. Balans, C. A., Advanced Engneerng Electromagnetcs, John Wley & ons, Mchalsk, K.A.; Mosg, J.R.;, "Multlayered meda Green's functons n ntegral equaton formulatons," Antennas and Propagaton, IEEE Transactons on, vol.45, no.3, pp , Mar Harrngton RF. Feld Computaton by Moment Methods. Macmllan, Kreger: New York, FL, Mosg J.R., Hall R.C., Gardol F.E., Numercal analyss of mcrostrp patch antennas. In Handbook of Mcrostrp Antennas, James JR, Hall P (eds). IEE-Peter Peregrnus: London, Mchalsk, K.A., Zheng, D., "Electromagnetc scatterng and radaton by surfaces of arbtrary shape n layered meda. I. Theory," Antennas and Propagaton, IEEE Transactons on, vol.38, no.3, pp , Mar Peterson, A.F., "Applcaton of the locally corrected Nyström method to the EFIE for the lnear dpole antenna," Antennas and Propagaton, IEEE Transactons on, vol.52, no.2, pp , Feb Tong M.., Chew W. C., "A hgher-order Nyström scheme for electromagnetc scatterng by arbtrarly shaped surfaces," Antennas and Wreless Propagaton Letters, IEEE, vol.4, no., pp , Mchalsk, K.;, "On the scalar potental of a pont charge assocated wth a tme-harmonc dpole n a layered medum," Antennas and Propagaton, IEEE Transactons on, vol.35, no., pp , Nov Polmerds A. G. and J.R. Mosg, J.R., "Complete sem-analytcal treatment of weakly sngular ntegrals on planar trangles va the drect evaluaton method", Int. J. Numercal Methods Eng., Vol. 83, pp , Durand, E., Électrostatque, Masson, 964 Fgure 9. urface currents on the patch. (a) Presented method and (b) Full MoM. CONCLUION In ths paper, a pont-wse approach that can be used effcently n the numercal soluton of Electrc Feld Integral Equatons has been presented. The algorthm s based on the so-called magc dstance concept whch defnes exactly the pont-to-pont equvalent of a four dmensonal ntegral. Magc dstance values are rgorously obtaned n the electrostatc case and generalzed to the electrodynamc case va the Mxed Potental formulaton of the Electrc Feld Integral Equaton. Wth the presented approach, pont-matchng schemas can now be also appled to the drect EFIE formulaton wthout need of castng t n terms of auxlary potentals. The advantages are manfold. On one sde, t s possble to deal drectly wth feld values, whch are physcal quanttes always unvocally defned. On the other hand, the procedure needed to obtan the MoM matrx s much smplfed and the computatonal effort s greatly reduced when compared wth a Galerkn formulaton, all ths wthout apprecable deteroraton n the overall accuracy. Work s n progress to extend these concepts to trangular meshes and more complex envronments requrng alternate defntons of the nvolved Green's functons. REFERENCE Internatonal Journal of RF and Mcrowave Computer-Aded Engneerng

8 BIOGRAPHIE Juan R. Mosg was born n Cadz, pan. He receved the Electrcal Engneer degree from the Unversdad Poltecnca de Madrd, Madrd, pan, n 973, and the Ph.D. degree from the Ecole Polytechnque Fédérale de Lausanne (EPFL), Lausanne, wtzerland, n 983. nce 99, he has been a Professor at EPFL and snce 2000, he has been the Head of the Laboratory of Electromagnetcs and Acoustcs (LEMA) at EPFL. In 984, he was a Vstng Research Assocate wth the Rochester Insttute of Technology, Rochester, NY, and yracuse Unversty, yracuse, NY. He has also held scentfc appontments wth the Unversty of Rennes, France, the Unversty of Nce, France, the Techncal Unversty of Denmark at Lyngby and the Unversty of Colorado at Boulder, UA. He s a also a Fellow of the IEEE, and a foundng member and actng Char of the European Assocaton&Conferences on Antennas and Propagaton (EurAAP and EuCAP). Dr. Mosg has authored fve chapters n books on planar antennas and crcuts and over 00 peer-revewed ournal papers. Hs current research nterests nclude electromagnetc theory, numercal methods, and planar antennas.